However, the LGW paradigm has been challenged recently by various examples in which the fluctuations from emergent degrees of freedom render the transition continuous.
The prime example is the transition between Neel and valence bond solid phase that separated by the deconfined QCP on the 2D square Heisenberg antiferromanets[1]}, [2]}. A large number of theoretical and numerical studies demonstrate that the deconfined QCP is continuous as a result of the emergent fractionalized "spinon" and noncompact U(1) gauge field[3]}, [4]}, [5]}, [6]}, [7]}.
Since the new degrees of freedom such as spinons emerge right at the QCP,
the LGW theory is fail to describe the deconfined QCP purely in terms of the space-time fluctuations of order parameter.
Another example that goes beyond the LGW picture is the fermion induced quantum critical point (FIQCP) which has attracted persistent attention in Dirac fermion systems
[8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}.
Evidences for FIQCP have been embodied in the transition from semimetal to \(\mathbb {Z}_3\)  Kekule valence-bond-solid (VBS) phase of (2+1)D fermions on the honeycomb lattice[8]}, [9]}, [18]}. The Kekule VBS pattern breaks the translational symmetry and breaks the continuous
\(U(1)\)  symmetry down to \(\mathbb {Z}_3\)[18]}, [36]}, [37]}, consequently, the Kekule phase transition allows a cubic term of VBS order parameter. From the view of point of LGW picture, the Kekule phase transition is expected to be first-order in the presence of cubic term of order parameter. Extensive studies, however, suggest that the presence of gapless fermion fluctuations at the critical point can dramatically change the nature of critical point and render a putatively first-order transition continuous[8]}, [18]}. Various theoretical methods have been applied to study the FIQCP, ranging from perturbative RG [10]}, [28]}, [29]}, [20]}
to functional RG[18]}, [12]}, [16]}, [47]}.
On the other hand, the sign-problem-free quantum Monte Carlo simulations for interacting fermions on lattice also push our
understanding of FIQCP [48]}, [49]}, [50]}, [51]}, [52]}.
